Learn German: Ich streite mit ihm. - I am arguing with him.

The preposition mit (with) always requires the dative case. Since you are arguing with "him" (a masculine pronoun), the accusative ihn becomes the dative ihm.

How would I ask "Who am I arguing with?" based on this sentence?

You would ask, "Mit wem streite ich?" When asking a question about a person involving a dative preposition like mit, you combine the preposition with wem, which is the dative form of wer (who).

Could I use the word damit to say "with him"?

No. Words that start with da- (like damit, darauf) are only used for objects or ideas, meaning "with it" or "on it." Because you are talking about a person here, you must keep the preposition and pronoun separate: mit ihm.

Does streiten mean a verbal argument or a physical fight?

The verb streiten specifically refers to a verbal argument, quarrel, or bickering. If you want to talk about a physical fight, you would use a different verb like kämpfen.